The diagonal mapping in mixed norm spaces

For any holomorphic function F in the unit polydisc Uⁿ of ℂⁿ, we consider its restriction to the diagonal, i.e., the function in the unit disc U of ℂ defined by 𝓓F(z) = F(z,...,z), and prove that the diagonal mapping 𝓓 maps the mixed norm space $H^{p,q,α}(Uⁿ)$ of the polydisc onto the mixed norm space $H^{p,q,|α|+(p/q+1)(n-1)}(U)$ of the unit disc for any 0 < p < ∞ and 0 < q ≤ ∞.